Let us start with the \((1+3)\) -dimensional de Sitter manifold, \(M\) , defined as a hyperboloid of radius \(\omega _{H}^{-1}\)  embedded in the five-dimensional flat space-time \(M^5\)  of coordinates \(z^A\)  (labeled by the indices \(A,\,B,...= 0,1,2,3,4\) ) and metric \(\eta ^5={\rm diag}(1,-1,-1,-1,-1)\)  [1]}, denoting by \(\omega _{H}\)  the de Sitter-Hubble constant (frequency). A local chart of coordinates \(x^{\mu }\)  (\(\mu ,\nu ,...=0,1,2,3\) ) can be introduced on \(M\)  giving the set of functions \(z^A(x)\)  which solve the hyperboloid equation, \(\eta ^5_{AB}z^A(x) z^B(x)=-\omega ^{-2}_{H}\) .
